Complex Path Integrals and the Space of Theories

Abstract

The Feynman Path Integral is extended in order to capture all solutions of a quantum field theory. This is done via a choice of appropriate integration cycles, parametrized by M in SL(2,C), i.e., the space of allowed integration cycles is related to certain Dp-branes and their properties, which can be further understood in terms of the "physical states" of another theory. We also look into representations of the Feynman Path Integral in terms of a Mellin-Barnes transform, bringing the singularity structure of the theory to the foreground. This implies that, as a sum over paths, we should consider more generic paths than just Brownian ones. Finally, we are able to study the Space of Theories through our examples in terms of their Quantum Phases and associated Stokes' Phenomena (wall-crossing).

51 pages, 33 figures (high definition versions available upon request), partial results were presented at 1st Winter Workshop on Non-Perturbative Quantum Field Theory, http://wwnpqft.inln.cnrs.fr/, and at Miami 2010, http://cgc.physics.miami.edu/ . v2: typos fixed, references added

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Cited by in corpus (10)

• Structure of Lefschetz thimbles in simple fermionic systems
• Some remarks on Lefschetz thimbles and complex Langevin dynamics
• The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painleve II Equation
• Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence
• Lefschetz thimbles and stochastic quantisation: Complex actions in the complex plane
• Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles
• Quantum field theories on the Lefschetz thimble
• Complex Classical Fields and Partial Wick Rotations
• A new look at the Helmholtz equation: Lefschetz thimbles and the einbein action
• Wronskians, dualities and FZZT-Cardy branes